To animate, I'm using the puppet tool>bones>structure. Everything is working great, but I now want to parent the hair artwork to the head artwork so that it moves when the head moves (the head is rigged to the structure etc.)

When I parent, it does not rotate when the head rotates (same problem with the eyes. Is this something that needs to be done BEFORE I puppet pin, add bones, add structure? Is there ANY way to do it now, rather than start all over again?

If there are bones on the puppet tool, they're controlling the layer, so parenting won't do anything (or would only create problems). As the bones will drag the layer, you don't have to parent it to anything.